The nonlinear control design problem for large angle reorientation manoeuvre of

spacecraft has a proper structure for the direct application of backstepping design as

its dynamics and kinematics are naturally in a cascade form. In this paper, the

robustness of the backstepping control against the uncertainties in the moment of

inertia matrix is investigated and a sufficient condition for the robust stability is

derived. Numerical simulations show the validity of the condition

Ali, I

Kim, J.K.

Radice, G

English

Enlighten

       Ali, I and Kim, J.K. and Radice, G (2008) Large angle reorientation manoeuvre of spacecraft using robust backstepping control. In: The 59th International Astronautical Congress, Glasgow, UK.   http://eprints.gla.ac.uk/4990/  Deposited on: 1 July 2009   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk  1 IAC-08-C1.8.1723  LARGE ANGLE REORIENTATION MANOEUVRE OF SPACECRAFT USING ROBUST BACKSTEPPING CONTROL   Imran Ali University of Glasgow, Glasgow, United Kingdom i.ali@aero.gla.ac.uk  Jongrae Kim University of Glasgow, Glasgow, United Kingdom jkim@aero.gla.ac.uk  Gianmarco Radice University of Glasgow, Glasgow, United Kingdom gradice@aero.gla.ac.uk    ABSTRACT  The nonlinear control design problem for large angle reorientation manoeuvre of spacecraft has a proper structure for the direct application of backstepping design as its dynamics and kinematics are naturally in a cascade form. In this paper, the robustness of the backstepping control against the uncertainties in the moment of inertia matrix is investigated and a sufficient condition for the robust stability is derived. Numerical simulations show the validity of the condition.    1. INTRODUCTION  BACKSTEPPING is a systematic Lyapunov-based control design method for nonlinear systems, especially those in a cascade form [ 1]. The basic idea is to use a part of the system states as virtual controls to control the other states. The name backstepping refers to the recursive nature of the control design procedure where a control law as well as a control Lyapunov function is recursively constructed to guarantee the stability. Generating a family of globally asymptotically stabilizing control laws is the main advantage of this method and it can be exploited for addressing robustness issues and solving adaptive problems.  Large angle reorientation manoeuvre problem of spacecraft, whose dynamics and kinematics are naturally in a cascade form, is a good candidate for this technique [ 2, 3]. However, the control actuators used for spacecraft attitude manoeuvre problems, e.g. reaction wheels, control moment gyros or thrusters, have an upper bound on the magnitude of torque they can exert onto the system and the simple or conventional backstepping control method may result in excessive control input beyond that saturation bound of the actuators.   2 A family of augmented Lyapunov functions proposed in [ 4] introduces a constant gain that is helpful to lower the required control torque bound. For this approach, the analytical bound for the control torque was derived in [ 5]. Lowering of the control torque bound may cause settling time performance degradation and [ 5] has also employed the nonlinear function based tracking function in [ 3] to reduce the settling time. The same conventional backstepping control law as considered by [ 5] is being analysed in the present paper for robustness against the uncertainties of the spacecraft moments of inertia.  The paper is organized as follows: Firstly, the kinematics and dynamics of rigid spacecraft attitude motion are summarised. Secondly, the analytical estimates of the moment of inertia matrix bounds in terms of eigenvalues and the sufficient condition of robust stability are presented. Finally, numerical simulation shows the validity of the robustness condition and the summary is presented.   2. RIGID SPACECRAFT ATTITUDE MOTION  Spacecraft is assumed to be a rigid body with actuators that provide torques about three mutually perpendicular axes that define a body-fixed frame with origin at the centre of mass of spacecraft. The equations of rotational motion of spacecraft are given by [ 6]   13 4 13 4 131 1( ),    2 2Tq q= − × = −q q qɺ ɺω ω ω  (1)  [ ]+ × =J J Tɺω ω ω  (2)  where 313 ∈ ℜq  and 4q ∈ ℜ  satisfy 213 13 4 1T q+ =q q , 13 4[ ,  ]T Tq=q q  denotes the unit quaternion that represents the orientation of spacecraft with respect to an inertial frame, 1 2 3[ , , ]Tω ω ω=ω  denotes the angular velocity of the spacecraft with respect to the inertial frame expressed in the body frame, T=J J  denotes the body frame referenced positive definite inertia matrix of spacecraft, 31 2 3[ , , ]TT T T= ∈ ℜT  denotes the control torque with components in the body frame. We define the three subscripts i, j and k as { }( , , ) (1, 2, 3),  (2, 3,1),  (3,1, 2)i j k ∈  and Eq. (1) can be written as   142( )i i k j j kq q q qω ω ω= − +ɺ  (3)  and choosing 1 2 3( , , )diag J J J=J  Eq. (2) becomes   i i j k ip uω ω ω= +ɺ  (4)  where ( ) /i j k ip J J J= −  and /i i iu T J= .   3. ROBUSTNESS ANALYSIS  In this section, the backstepping controller design procedure is summarised and the sufficient condition for the robustness with respect to the moment inertia uncertainties is derived.  The candidate Lyapunov function for the kinematics subsystem stabilization is   2 2113 42(1 )V q= + −  q  (5)   3 The pseudo control law for the kinematics subsystem stabilization sω  is written as   4 13sgn( ) ( )s s q= − qω φ  (6)  where s is a positive constant and [ ]13 1 2 3( ) ( ) ( ) ( ) Tq q qφ φ φ= =qφ φ  with the nonlinear tracking function ( )iqφ  as given in [ 3]   1( ) tan ( )i iq qφ α β−=  (7)  where α  and β  are positive constants. The function sgn(.) denotes the sign function defined  by [3]   1,      0sgn( )1,        0or1,      0sgn( )1,        0xxxxxx− <=≥− ≤=> (8)  The Lyapunov function in Eq. (5) is augmented for the overall system as follows [ 4]:   2 221 113 42 2(1 )sU q= + − + −  q Ω Ω  (9)  where [ ]1 2 3( ) ( ) ( ) ( ) TΩ ω Ω ω Ω ω= =Ω Ω ω ,[ ]1 2 3( ) ( ) ( ) ( ) Ts s s s sΩ ω Ω ω Ω ω= =Ω Ω ω , and ( )Ω ⋅  is the class κ∞ function, i.e. it is zero at zero, strictly increasing and becomes unbounded as the argument is unbounded [ 4]. The time derivative of Eq. (9) becomes   ( ) ( )( ) ( )( ) ( )14 1321 14 13 4 132 2113 13214 1321sgn( )sgn( ) sgn( )(diag diag( )( sgn( )diag( diag diag diagTTs sT Ts sTs s sTTss s s sU qq qsq−= + − −= + − +′ ′− −= − +− +′ ′− − −qq ) qq q) q)ɺ ɺ ɺɺ ɺɺ ɺω Ω Ω Ω Ωω ω ωΩ Ω (Ω )ω (Ω )ωΦω ωω ω Ω Ω (Ω )ω (Ω )ω (10)  where [ ]1 2 3( ) ( ) ( ) ( ) TΩ ω Ω ω Ω ω′ ′ ′ ′ ′= =Ω Ω ω ,[ ]1 2 3( ) ( ) ( ) ( ) Ts s s s sΩ ω Ω ω Ω ω′ ′ ′ ′ ′= =Ω Ω ω , and ( )xΩ ′  defines the derivative of ( )xΩ  with respect to x. Moreover, for a vector [ ]1 2 3, , Ta a a=a , diag( )a  denotes the diagonal matrix 1 2 3diag( , , )a a a . In order to make the time derivation of the Lyapunov function be negative definite, the control law is chosen as follows:  (( ) ]1 1 14 1321diag( ) diag( ) diag( ) sgn( )( ) diag( )s ss s sqg− −− ×′= − − − +′− − +u qJ JɺΩ Ω Ω ω ωω ω Ω ω ω ω (11)  Substituting the above backstepping controller in Eq. (10) we get   113 132( ) ( (T Ts sU s g= − − − −q q ) )ɺ φ ω ω ω ω  (12)  where g is a positive constant. Hence, the control law of Eq. (11) is globally asymptotically stable. Let  4 aJ  denote the actual spacecraft inertia matrix which is different from J , the one assumed for the design of the control law of Eq. (11). Substituting the control input of Eq. (11) in conjunction with the actual inertia matrix aJ , Eq. (10) becomes ( )113 1321 11( ) ( ( )( diag( diag diagT Ts sTs s s a aU s g− − ×− ×= − − − − −′− − − −q q )) ) J JJ Jɺ φ ω ω ω ωω ω ω ω Ω Ω (Ω ) ωω ω(13)  Considering the simple case of ( )i iω ηωΩ =  with 1η >  and defining s= −e ω ω , 1a δ= +J J J  and 1 12a δ− −+J J J=  where ( )1 12 3 1δ δ − −= − +J I J J J , [ 7], Jλ≤J , 1 1 / Jλ− ≤J , 1 1δ γ≤J  and 2 2δ γ≤J  the above equation can be written as   ( )( )( )2113 1322 11 2 2 1113 13222 11 2 2 1113 132221 2 1 2( )( )( )TTTTJ JU s gs gs gη δ δ δ δη δ δ δ δη γ λ γ λ γ γ− × × ×−= − − −+ +≤ − − −+ +≤ − − −+ +q q ee J J J J J Jq q e eJ J J J J Jq q e eɺ φω ω ω ωφωφω(14)  As a result, the negative definiteness of Uɺ is guaranteed if the following inequality is satisfied:   ( ) 22 1 2 1 2J Jg η γ λ γ λ γ γ≥ + +e ω  (15)  and the whole system is robustly stable with respect to the moment of inertia uncertainties.  4. NUMERICAL SIMULATION  The usefulness of the condition given by Eq. (15) is demonstrated through the numerical simulation of a benchmark rest-to-rest slew manoeuvre [ 2, 3, 5]. The simulation scenario is as follows:   21 2 30diag( , , ) diag(10,15, 20)       (kg m )( ) [0.4646  0.1928  0.8047  0.3153]( ) [0  0  0  1]TTfJ J Jtt= ===Jqq where 0t  and ft  are the starting and the final times, respectively. The values for the control gain g and α  and β  for the nonlinear tracking function ( )iqφ  are chosen the same as given in [ 3], i.e., 10, 0.75 and 8.0, respectively. Also, the same true moment of inertia matrix in [ 3], aJ , is adopted as follows:  2diag(8,16.5, 24)       (kg m )a =J  For the given J  and aJ , the parameters for the bound, Eq. (15), are 20Jλ = , 10Jλ = , 1 4γ =  and 2 0.025γ = . Finally, selecting the control gains η  and s equal to 6 and 10, respectively, the sufficient condition for the robust stability given by Eq. (15) becomes 210≥e ω . Fig. (1)–(5) show the simulation results for the specified manoeuvre. The whole scenario is the same as considered in [ 5] except the control gains η  and s have been retuned in order to ensure the robustness against the uncertainties in the moments of inertia. The peak control torque from Fig. 3 is similar to the one reported in the simulation results of [ 5] but the performance in terms of the settling time has been sacrificed because of the effort of meeting the robustness condition, Eq. (15). A considerate retuning of all of control gains may rectify this issue of the sluggish motion response.  5   0 10 20 30 40-0.8-0.6-0.4-0.20time [sec]Angular velocity [rad/sec]  ω1ω2ω3  Fig. 1. Angular velocity ω  history  0 10 20 30 4000.51time [sec]Attitude quaternion  q1q2q3q4  Fig. 2. Unit attitude quaternion q  history.  0 10 20 30 40-10-5051015time [sec]Control torque [Nm]  T1T2T3 Fig. 3. Control torque Τ  history. 0 10 20 30 400246time [sec]Tracking error  |e1||e2||e3|  Fig. 4. Tracking error e  history.  0 10 20 30 400510time [sec]Inertia robustness  10||ω||2||e||  Fig. 5. Inertia robustness condition   5. CONCLUSION  The conventional backstepping controller for the spacecraft large angle reorientation manoeuvre problem has been analysed in the context of robustness against the uncertain variations in the moments of inertia and a sufficient condition has been derived that ensures the robust stability to the prescribed bounded uncertainties.   REFERENCES  1. Khalil, H. K., Nonlinear Systems, 3rd ed., Prentice Hall, Upper Saddle River, NJ., 2002, Chap. 14.  6 2. Krstic, M., and Tsiotras, P., “Inverse Optimal Stabilization of a Rigid Spacecraft,” IEEE Transactions on Automatic Control, Vol. 44, No. 5, 1999, pp. 1042–1050. 3. Kim, K.-S., and Kim, Y., “Robust Backstepping Control for Slew Maneuver Using Nonlinear Tracking Function,” IEEE Transactions on Control Systems Technology, Vol. 11, No. 6, 2003, pp. 822–829. 4. Mazenc, F., and Iggidr, A., “Backstepping with Bounded Feedbacks,” Systems and Control Letters, Vol. 51, 2004, pp. 235–245. 5. Ali, I., Radice, G. M., and Kim, J., “Analytical Control Torque Bound for Backstepping Control of Spacecraft Attitude Maneuver,” preparing to submit to Journal of Guidance, Control and Dynamics. 6. Shuster, M. D., “A Survey of Attitude Representations,” Journal of the Astronautical Sciences, Vol. 41, No. 4, 1993, pp. 439–517. 7. Kim, J. and Crassidis, J. L., “Spacecraft Attitude Control Using Approximate Receding Horizon Model-Error Control Synthesis”, Journal of Guidance, Control and Dynamics, Vol. 29, No. 5,  2006, pp. 1023-1031 

A Survey of Attitude Representations,”

Analytical Control Torque Bound for Backstepping Control of Spacecraft Attitude Maneuver,” preparing to submit to Journal of Guidance, Control and Dynamics.

Backstepping with Bounded Feedbacks,”

Inverse Optimal Stabilization of a Rigid Spacecraft,”

Nonlinear Systems, 3rd ed., Prentice Hall, Upper Saddle River,

Robust Backstepping Control for Slew Maneuver Using Nonlinear Tracking Function,”

Spacecraft Attitude Control Using Approximate Receding Horizon Model-Error Control Synthesis”,

Large angle reorientation manoeuvre of spacecraft using robust backstepping control

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Large angle reorientation manoeuvre of spacecraft using robust backstepping control

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